If new super-powers normally go haywire in a highly dangerous manner, laws will no doubt require their isolation from society until they master them. Or even a sizable percentage -- governments will insist, though supers will be more likely to resist. (Especially the overconfident and mistrustful ones.) Whether a super can successfully gauge the danger will have deep impacts; a character who hides powers to avoid the isolation will be an outlaw if it comes out, so the likelihood of its happening will increase the pressure to accept the fate.
Slowly growing powers will decrease the danger -- both to society, and to forcing the super into the training.
The question of how long it will take it train will change much about their lives. The other burning question is whether they need also to defend themselves. If a person who can fly can just learn to avoid doing so in moments of high emotion, that's one thing. If he has to learn self-defense and more to avoid being pressured into a life of crime, he may find that it's impossible to do anything but train. Highly powerful supers will at least minimize the risk of normal humans being able to pressure them; they can even protect their families more, though that can be ticklish. Governmental offers of a specially protected residence may gravely reduce the need for coercion. (Throwing together lots of young superheroes in hopes they will fall in love and thus reduce the need to protect normal spouses is a nice side-effect.) But nothing prevents other super-powers from looking for people to serve as henchmen, or cannon fodder. Discovering you have a superpower may mean the death of dreams.